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CISCEISC (Science) ISC Class 11
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अवधारणा स्पष्टीकरण और वीडियो२०९
पाठ्यक्रम

Motion in Two Dimensions-Motion in a Plane - Relative Velocity in Two Dimensions

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Topics

  • Introduction
  • Formula: Velocity of A relative to B
  • Formula: Velocity of B relative to A
  • Characteristics
  • Chaining Relative Velocities
  • Significance
  • Example
  • Real-Life Example
Maharashtra State Board: Class 11

Introduction

  • Relative velocity is important to compare the motion of two moving objects.
  • It shows how fast one object appears to move when seen from another moving object.
  • This concept works for objects moving in a straight line and in a plane.
  • Knowing relative velocity helps in solving problems involving two or more moving bodies, like vehicles, planes, or boats.
Maharashtra State Board: Class 11

Formula: Velocity of A relative to B

\[\vec{v}_{AB}=\vec{v}_A-\vec{v}_B\]

where:

  • \[\vec v_A\] = Velocity of object A
  • \[\vec v_B\] = Velocity of object B
  • \[\vec v_{AB}\] = Velocity of A relative to B
  • The subscripts indicate the order: for \[\vec v_{AB}\], A is the observer, B is referenced.
Maharashtra State Board: Class 11

Formula: Velocity of B relative to A

\[\vec{v}_{BA}=\vec{v}_B-\vec{v}_A\]

where:

  • \[\vec v_A\] = Velocity of object A
  • \[\vec v_B\] = Velocity of object B
  • \[\vec v_{AB}\] = Velocity of A relative to B
  • The subscripts indicate the order: for \[\vec v_{BA}\], B is the observer, A is referenced.
Maharashtra State Board: Class 11

Characteristics

  • The magnitude of \[\vec{\mathrm{v}}_{AB}\] and \[\vec{\mathrm{v}}_{BA}\] are equal.
  • The direction of \[\vec{\mathrm{v}}_{AB}\] and \[\vec{\mathrm{v}}_{BA}\] are opposite.
  • For multiple objects, relative velocities can be added sequentially, maintaining the order of subscripts.
Maharashtra State Board: Class 11

Chaining Relative Velocities

To find the relative velocity between two objects:

  • Subtract the velocity vector of the reference object from the velocity vector of the observed object.

When multiple objects are involved, relative velocities can be chain-added, following the order of their movement (A→B→C→...→Z).

Maharashtra State Board: Class 11

Significance

  • Helps compare how fast one object appears from another moving object.
  • Important for solving problems in navigation, aviation, and vehicle movement.
  • Useful in analyzing accidents and collisions.
Maharashtra State Board: Class 11

Example

An aeroplane is travelling northward with a speed of 300 km/hr with respect to the Earth. Wind blows from east to west at 100 km/hr. What is the velocity of the aeroplane with respect to the wind?

Solution Steps:

  • Let velocity of the aeroplane w.r.t. Earth:\[\vec{v}_{AE}\]

  • Let velocity of wind w.r.t. Earth: \[\vec{v}_{WE}\]

  • The velocity of the aeroplane w.r.t. wind:
    \[\vec{v}_{AW}=\vec{v}_{AE}-\vec{v}_{WE}\]

  • Express in vector form:
    Since north is +y and east is +x,
    \[\vec{v}_{AW}=100\hat{i}+300\hat{j}\]

  • Magnitude:
    \[\sqrt{(100)^{2}+(300)^{2}}=100\sqrt{10}\mathrm{km/hr}\]

  • Direction:
    \[\theta=\tan^{-1}(\frac{300}{100})=71.6^\circ\text{north of east}\]

Maharashtra State Board: Class 11

Real-Life Example

  • A person walking on a moving train: The speed observed by someone on the ground and in the train looks different due to relative velocity.
  • Boats crossing rivers: The effective speed and direction depend on both boat speed and river current.
  • Airplanes flying in windy conditions: The net motion is a combination of plane speed and wind speed.

Test Yourself

Shaalaa.com | Motion in plane part 21 (Relative velocity)

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Related QuestionsVIEW ALL [4]

A clock has a continuously moving second's hand of 0.1 m length. The average acceleration of the tip of the hand (in units of ms-2) is of the order of ______. 

A swimmer wants to cross a river from point A to point B. Line AB makes an angle of 30° with the flow of river. Magnitude of velocity of the swimmer is same as that of the river. The angle θ with the line AB should be ______°, so that the swimmer reaches point B. 

A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 km/h passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 m s–1 to hit the plane? At what minimum altitude should the pilot fly the plane to avoid being hit? (Take g = 10 m s–2).

Rain is falling vertically with a speed of 30 m s–1. A woman rides a bicycle with a speed of 10 m s–1in the north to south direction. What is the direction in which she should hold her umbrella?

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